# Fourier Transform- is this right?

1. Apr 15, 2007

### FrogPad

If someone wouldn't mind checking if this answer is correct, that would be awesome.

$$x(t) =$$ a triangle with points (-2,0), (0,1), and (2,0).

I am supposed to compute the Fourier transform ($\hat x (\omega)$) of $x(t)$.

"Solution"
$$m = \frac{y_2-y_1}{x_2-x_1}=\frac{1}{2}$$

Thus, $$\frac{dx(t)}{dt} =$$ a pulse with height $\frac{1}{2} [/tex] from (-2 to 0), and a pulse with height [itex] \frac{-1}{2}$ from (0 to 2).

Let $$k(t) = \frac{dx(t)}{dt}$$

Since, $$x(t) = \int_{-\infty}^x k(t)dt \leftrightarrow \frac{1}{j\omega} \hat k(\omega) + \hat k(0) \delta(\omega)$$

Let $$x'(t) =$$ a pulse from -1 to 1 with height 1

Then, $$\frac{1}{2} x'(t) \leftrightarrow \frac{\sin \omega}{\omega}$$

and, $$k(t) = \frac{1}{2}x'(t+1) - \frac{1}{2}x'(t-1)$$

Thus, $$\hat k(\omega) = e^{j\omega}\frac{\sin \omega}{\omega} - e^{-j\omega}\frac{\sin \omega}{\omega}$$

$$\hat x (\omega) = \frac{2 \sin_c \omega}{\omega} \left( \frac{e^{j\omega}-e^{-j\omega}}{2j} \left) + 2 \sin_c(0)\cos(0)\delta(\omega) = 2 ( \sin_c^2 \omega + \delta(\omega))$$

note: $$\sin_c(x) = \frac{sin x}{x}$$ (I don't know how to write the sinc function)
Does this look right?

Thanks

Last edited: Apr 15, 2007
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